Thermomechanical contact of viscoelastic bodies is a non-linear time- and temperature-dependent problem. Consideration of temperature as an independent variable destroys the convolution integral form of the viscoelasticity constitutive relations. This paper presents a computational procedure capable of predicting the quasistatic response of uncoupled, thermoviscoelastic, frictionless contact problems. The contact problem as a convex programming model is solved throughout an incremental procedure. The Wiechert model is adopted to simulate the linear behaviour of viscoelastic materials. The temperature dependency of viscoelasticity is accounted for by applying the time—temperature superposition principle, in which the William, Landel, and Ferry relationship is adopted to determine the shift factor. Thus, the constitutive equations are transformed to be a function of the reduced time as the only independent variable, maintaining the convolution integral form. Therefore, the complications that arise during the direct integration of these equations, especially with contact problems, are avoided. Two different illustrative examples are included to demonstrate the applicability of the proposed procedure.
LeadermanH.Elastic and creep properties of filamentous materials and other polymers, 1943, pp. 175–185 (Textile Foundation, Washington, DC).
2.
TobolskyA. V.Stress relaxation studies of the viscoelastic properties of polymers. J. Appl. Phys., 1956, 27(7), 673–685.
3.
SchwarzlF.StavermanA. J.Time-temperature dependence of linear viscoelastic behavior. J. Appl. Phys., 1952, 23(8), 838–843.
4.
ChazalC.ArfaouiM.Further development in thermodynamics approach for thermoviscoelastic materials. Mechanics of Time-Dependent Materials, 2001, 5, 177–198.
5.
HiltonH. H.YiS.Anisotropic viscoelastic finite element analysis of mechanically and hygrothermally loaded composites. U-C, Technical Report, AAE 90–8, UILU ENG 90–0508, University of Illinois, 1990.
6.
YiS.AhmedM. F.RameshA.Data parallel computation for thermo-viscoelastic analysis of composite structures. Adv. Eng. Software, 1996, 27, 97–102.
7.
ZocherM. A.A Thermoviscoelastic Finite Element Formulation for the Analysis of Composites. PhD Dissertation, Texas, A & M University, Collage Station, TX, 1995.
8.
ZocherM. A.GrovesS. E.AllenD. H.A three-dimensional finite element formulation for thermoviscoelastic orthotropic media. Int. J. Numer. Meth. Engng, 1997, 40, 2267–2288.
9.
PoonH.AhmedH. A.A material point time integration procedure for anisotropic, thermo-reheologically simple viscoelastic solids. Comput. Mech., 1998, 21, 236–242.
10.
ParkS. W.KimY. R.Analysis of layered viscoelastic system with transient temperatures. J. Eng. Mech., 1998, 124(2), 223–231.
11.
BonettiE.BonfantiG.Existence and uniqueness of the solution to a 3D thermoviscoelastic system. Electron. J. Differential Equations, 2003, 2003(50), 1–15.
12.
BatraR. C.LevinsonM.BetzE.Rubber covered rolls — the thermoviscoelastic problem; a finite element solution. Int. J. Numer. Meth. Engng, 1976, 10, 767–785.
13.
BatraR. C.Cold sheet rolling, the thermoviscoelastic problem, a numerical solution. Int. J. Numer. Meth. Engng, 1977, 11, 671–682.
14.
ChenW. H.ChangC. M.YehJ. T.Finite element analysis of viscoelastic contact problems with friction. In Proceedings of the Fifteenth National Conference on Theoretical and Applied Mechanics, Tainan, Taiwan, Republic of China, 1991, pp. 713–720.
NaghiehG. R.JinZ. M.RahnejatH.Characteristics of frictionless contact of bonded elastic and viscoelastic layered solides. Wear, 1999, 232, 243–249.
17.
RochdiM.ShillorM.SofoneaM.Quasi-static viscoelastic contact with normal compliance and friction. J. Elasticity, 1998, 51, 105–126.
18.
ShillorM.SofoneaM.A quasistatic viscoelastic contact problem with friction. Int. J. Eng. Sci., 2000, 38, 1517–1533.
19.
AwabiB.RochdiM.SofoneaM.Abstract evolution equations for viscoelastic frictional contact problems. J. Appl. Math. Phys., 2000, 50, 1–18.
20.
HanW.SofoneaM.Evolutionary variational inequalities arising in frictional contact problems. SIAM J. Numer. Anal., 2000, 38, 556–579.
21.
AwabiB.ChauO.SofoneaM.Variational and numerical analysis for a frictional contact problem for viscoelastic bodies. Int. Math. J., 2002, 1, 333–348.
22.
HanW.SofoneaM.Quasistatic contact problems in viscoelasticity and viscoplasticity, ‘Studies in Advanced Mathematics’ series, 2002 (American Mathematical Society, International Press, Somerville, MA, pp. 209–285.
23.
FernandezJ. R.HanW.SofoneaM.Numerical simulations in the study of frictionless contact problems. Int. J. Appl. Math. Comp. Sci., 2003, 30, 97–105.
24.
MahmoudF. F.El-ShafeiA. G.AttiaM. A.An incremental adaptive procedure for viscoelastic contact problems. Int. J. Tribol., 2007, 129(2), 305–313.
ChangC. M.ChenW. H.Thermoviscoelastic contact analysis with friction by an incremental thermal relaxation procedure. Comput. Meth. Appl. Mech. Engng, 1996, 130, 151–162.
27.
AndrewsK. T.ShillorM.WrightS.KlarbringA.A dynamic thermoviscoelastic contact problem with friction and wear. Int. J. Eng. Sci., 1997, 35(14), 1291–1309.
28.
NakaoM.RiveraJ. E. M.The contact problem in thermoviscoelastic materials. J. Math. Anal. Appl., 2001, 264, 522–545.
29.
CopettiM. I. M.FrenchD. A.Numerical solution of a thermoviscoelastic contact problem by a penalty method. SIAM J. Numer. Anal., 2003, 41(4), 1487–1504.
30.
CopettiM. I. M.Numerical approximation of a thermoviscoelastic problem to the contact of two rods by a penalty method. Numer. Meth. Part. Differ. Equat., 2004, 20, 481–493.
31.
CopettiM. I. M.Error analysis for a finite element approximation of a thermoviscoelastic contact problem. J. Comput. Appl. Math., 2005, 180, 181–190.
32.
MachirajuC.PhanA. V.PearsallA. W.MadanagopalS.Viscoelastic studies of human subscapularis tendon: Relaxation test and a wiechert model. Comput. Meth. Programs Biomed., 2006, 83, 29–33.
33.
WilliamM.LandelR.FerryJ.The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. J. Am. Chem. Soc., 1955, 77(14), 3701–3707.
34.
DundersJ.Properties of Elastic Bodies in Contact. In The Mechanics of Contact Between Deformable Bodies (de PaterA. D.KalkerJ. J. Eds), 1975, pp. 54–66 (Delft University Press, The Netherlands).
35.
SchaperyR. A.Viscoelastic behavior and analysis of composite materials. In Mechanics of Composite Materials (SendeckyjG. P. Ed.), 1974, pp. 85–168 (Academic Press, New York).
36.
BairS.JarzynskiJ.WinerW. O.The temperature, pressure and time dependence of lubricant viscosity. Int. J. Tribol., 2001, 34(7), 461–468.
37.
FanB.KazmerD.Effect of low temperature shift factor modeling on predicted part quality. In ANTEC Conference Proceedings, 2003, 1, pp. 938–942.
38.
NarhiW.MehtaS.Application of time-temperature superposition to stress relaxation in elastomers. In ANTEC Conference Proceedings, 2003, 3, pp. 3014–3021.
39.
TauzowskiP.KleiberM.Parameter and shape sensitivity of thermo-viscoelastic response. Comput. Struct., 2006, 84, 385–399.
40.
MahmoudF. F.Al-SaffarA. K.El-HadiA. M.Solution of the non-conformal unbonded contact problems by the incremental convex programming method. Comput. Struct., 1991, 35(1/2), 1–8.
41.
HassanM. M.MahmoudF. F.A generalized adaptive incremental approach for solving inequality problems of convex nature. Struct. Eng. Mech., 2005, 18(4), 461–474.
42.
FletcherR.Practical Methods of Optimization., 1987 (Wiley, New York), pp. 196–219.
43.
BatheK. J.Finite Element Procedures, 1996 (Prentice-Hall, Inc., Englewood Cliffs, New Jersey), pp. 461–484.
44.
WriggersP.SimoJ. C.TaylorR. L.Penalty and augmented lagrangian formulation for contact problems. In Proceedings of the NUMETA1985 Conference, Swansea, 7–11 January, 1985.
45.
StupkiewiczS.Extension of the node-to-segment contact element for surface-expansion-dependent contact laws. Int. J. Numer. Meth. Engng, 2001, 50, 739–759.
46.
LeeE. H.RadokR. M.The contact problem for viscoelastic bodies. J. Appl. Mech., 1960, 27(3), 438–444.
47.
HunterS. C.The Hertz problem for a rigid spherical indenter and a viscoelastic half-space. J. Mech. Phys., 1960, 8, 219–234.
48.
YangW. H.The contact problem for viscoelastic bodies. J. Appl. Mech., 1966, 33(2), 395–401.
49.
TingT. C. T.The contact stresses between a rigid indenter and a viscoelastic half-space. J. Appl. Mech., 1966, 33(4), 845–854.
50.
TingT. C. T.Contact problems in the linear theory of viscoelasticity. J. Appl. Mech., 1968, 35(4), 248–254.
